This would mean that for the first time in modern Spanish history, the Socialist party (PSOE) would be demoted to the rank of third party in the country - replaced as the main left-wing force by upstart Podemos and its allies.
The results published by research firm CIS said that a coalition recently formed by Podemos and green-communists Izquierda Unida - as well as its far-left allies - would get 25.6 percent of the votes in the June 26th elections
The Socialist party, according to CIS, would only garner 21.2 percent, which would mean at least 10 fewer parliamentary seats than the so-called Unidos Podemos coalition and its allies.
"Four points, that's a lot," said Belén Barreiro, a sociologist who heads up MyWord, another polling firm.
"There can be variations in the campaign but it will be very difficult for the PSOE to manage to overtake Podemos again."
According to the CIS report, the ruling, conservative Popular Party (PP) would come first with 29.2 percent of the vote and at least 118 seats.
The predicted result would mean that no party will get an absolute majority in parliament, as was the case in December elections.
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They will be forced back to the negotiating table - as was also the case after the last polls - but under immense pressure to reach a coalition deal this time rather than go to a third round of elections.
And the Socialists, if they come third as currently predicted, will be stuck between a rock and a hard place.
The PSOE could choose to pact with the PP, its long-standing arch-rival, or with the far-left coalition led by its other, newer arch-rival Podemos.
"If it comes third the PSOE only has two options," a high-ranking Socialist party member told AFP on condition of anonymity.
"Dropping a nuclear bomb on the European Union (supporting the Podemos-led coalition) or dropping it on the PSOE (supporting the PP)."
Analysts say that either way, coming third will deal a huge blow to the Socialists.
"That will... probably tear the party apart," said Pablo Simón, politics professor at Madrid's Carlos III University.